Multiple choice

If one root of the equation, $ (5k)x^{2}+13x+(2k^2-k)=0 $, is the reciprocal of the other, then

  1. $k = 0$
  2. $k = -\displaystyle\frac{1}{2}$
  3. $k =3 $
  4. $ k = \displaystyle \frac{-5}{2}$
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C Correct answer
Explanation

If one root is the reciprocal of the other, the product of roots (c/a) must be 1. Here, (2k^2 - k) / (5k) = 1. Assuming k is not 0, 2k^2 - k = 5k, so 2k^2 - 6k = 0, which gives 2k(k-3) = 0. Thus k=3.

AI explanation

If the roots are reciprocals of each other, their product must be equal to 1. For a quadratic equation, the product of the roots is given by c divided by a. Substituting the given coefficients results in (2k^2 - k) divided by 5k equaling 1. Solving this equation yields 2k^2 - 6k = 0, which factors to 2k(k - 3) = 0. Excluding k = 0 to keep the equation quadratic, the valid solution is k = 3.

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